[noqc]

I have a silly question, and I'm going to shamelessly use HN to ask it.

In Kitaev's construction of the high purity approximation to a magic state, he starts with the assumption that we start with a state which can be represented as the tensor product of n mixed states which are "close enough". I don't understand where this separability property comes from. My (very) naive assumption would be that there is some big joint state which you have a piece of, and the information that I have about this piece are n of its partial traces, which are indeed n copies of the "poor man's" magic state.

Can I know more than that? There's lots of stuff in the preimage of these partial traces. Why am I allowed to assert that I have the nicest one?

[Strilanc]

Distillation will still work if the inputs are slightly entangled with each other or with other qubits.

I recommend just simulating the specific case you're worried about. It's only a 15 qubit circuit; not at all expensive to check. You'll either see it working and stop worrying, or have an amazing concrete counter example to publish.

[ziofill]

Can it be that he assumes you have some device that produces somewhat bad magic states and then you distill them into a better one? That would be the typical situation in practice.